Stability of the Discretized Pantograph Differential Equation

In this paper we study discretizations of the general pantograph equation y'(t) = ay(t) + by(6(t)) + cy'((t)), f>0, y(0)=y0, where a , b , c , and yo are complex numbers and where 9 and <¡> are strictly increasing functions on the nonnegative reals with 0(0) = <^>(0) = 0 and 8(t) < t, 4>(t) < t for positive /. Our purpose is an analysis of the stability of the numerical solution with trapezoidal rule discretizations, and we will identify conditions on a , b , c and the stepsize which imply that the solution sequence {yn/^o 's DOunded or that it tends to zero algebraically, as a negative power of n . X. Introduction In this article we shall consider, in the most general form, the generalized pantograph equation (1.1) y'(t) = ay(t) + by(d(t)) + cy'(0, y(0)=y0, where a, b, c, and yo are complex and where 8 and (t)) is called the "neutral" term. Our principal focus in this paper is the case of proportional delays 8(t) = qt and 4>(t) = pt, where q and p are between 0 and 1. The equation (1.1) can also be considered for vector-valued y and matrices a, b, c, but we will dispense with this generalization here. There are many applications for the generalized pantograph equation. Here we only mention applications in number theory (Mahler [15]), in electrodynamics (Fox et al. [9]) and the collection of current by the pantograph of an electric locomotive (whence its name; cf. Ockendon and Tayler [18]), and in nonlinear dynamical systems (Derfel [6]). A more comprehensive list features in Iserles [12]. Delay differential equations with constant delays, i.e., c = 0 and 6(t) = t-x, where one also prescribes y's values on (-t, 0), have been investigated extensively in the past (see, for instance, Bellman and Cooke [1,2] and Hale [10]). Received by the editor October 18, 1991 and, in revised form, May 8, 1992. 1991 Mathematics Subject Classification. Primary 34K20, 65L20; Secondary 34K40. © 1993 American Mathematical Society 0025-5718/93 $1.00+ $.25 per page